Superfluid Hydrodynamics Near the λ-Point 22

In the theory of second-order phase transitions one usually introduces a small parameter, describing how close the state of the system is to the λ-point. The thermodynamic potentials are then expanded in a series in this parameter whose value is determined by minimizing the potential. Far from the λ-point we are dealing with an ideal gas of excitations and the normal density ρ _n (and consequently also the superfluid density ρ _S) can be calculated by using formula 2-22. Near the λ-point such an approach is not possible and one must use a method of expansion in a small parameter. The density ρ _S may be used as such a parameter, since it vanishes at the λ-point. It turns out to be more convenient to introduce a complex function ψ(x,y,z,t) = ηe^iφ defined in such a way that (17-1) ρ s =   m | ψ | 2 v s = ℏ m ∇ φ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429502897/20d72e0b-9d39-4175-acc6-a943547edf8a/content/eq507.tif"/>